Binary Relations and Equivalence Relations, Exercises of Economics

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Binary Relations

Part One

Outline for Today ● Binary Relations ● Reasoning about connections between objects. ● Equivalence Relations ● Reasoning about clusters. ● A Fundamental Theorem ● How do we know we have the “right” defnition for something?

What exactly is a binary relation?

a b

aRb R

Binary Relations ● A binary relation over a set A is a predicate R that can be applied to pairs of elements drawn from A. ● If R is a binary relation over A and it holds for the pair ( a , b ), we write aRb.

3 = 3 5 < 7 Ø ⊆ ℕ

● If R is a binary relation over A and it does not hold for the pair ( a , b ), we write aR̸b.

4 ≠ 3 4 <≮ 3 ℕ ⊆≮ Ø

Properties of Relations ● Generally speaking, if R is a binary relation over a set A , the order of the operands is signifcant. ● For example, 3 < 5, but 5 <≮ 3. ● (^) In some relations order is irrelevant; more on that later. ● Relations are always defned relative to some underlying set. ● It's not meaningful to ask whether ☺ ⊆ 15, for example, since ⊆ is defned over sets, not arbitrary objects.

Visualizing Relations ● (^) We can visualize a binary relation R over a set A by drawing the elements of A and drawing a line between an element a and an element b if aRb is true. ● (^) Example: the relation a ≠ b over the set {1, 2, 3, 4} looks like this:

Visualizing Relations ● (^) We can visualize a binary relation R over a set A by drawing the elements of A and drawing a line between an element a and an element b if aRb is true. ● (^) Example: the relation a = b over the set {1, 2, 3, 4} looks like this:

Answer at PollEv.com/cs103 or text CS103 to 22333 once to join, then A , B , C , D , E , or F. Answer at PollEv.com/cs103 or text CS103 to 22333 once to join, then A , B , C , D , E , or F. Below is a picture of a binary relation R over the set {1, 2, …, 8}. Which of the following is a correct defnition of the relation R? A. xRy if x = 3 and y = 5 B. xRy if y = x + 2 C. yRx if y = x + 2 D. R = + E. None of these F. More than one of these Below is a picture of a binary relation R over the set {1, 2, …, 8}. Which of the following is a correct defnition of the relation R? A. xRy if x = 3 and y = 5 B. xRy if y = x + 2 C. yRx if y = x + 2 D. R = + E. None of these F. More than one of these

Capturing Structure

Partitions

Partitions and Clustering ● If you have a set of data, you can often learn something from the data by fnding a “good” partition of that data and inspecting the partitions. ● Usually, the term clustering is used in data analysis rather than partitioning. ● Interested to learn more? Take CS161 or CS246!

What's the connection between partitions and binary relations?